Scalable Quantum Simulations of Scattering in Scalar Field Theory on 120 Qubits

TL;DR

This work addresses the challenge of real-time scattering in quantum field theory by developing physics-informed scalable variational circuits (SVC) that compress vacuum preparation, wavepacket initialization, and time evolution for a 1D lattice scalar field theory. By combining SC-ADAPT-VQE and brickwall variational layers, the authors construct shallow, scalable circuits whose parameters are determined classically on small systems and extrapolated to larger sizes, enabling large-scale simulations with limited quantum resources. An error-mitigation pipeline (DD, Pauli twirling, TREX, and ODR) enables extraction of meaningful observables from circuits with up to 4924 two-qubit gates and depth 103, with results on 120 qubits qualitatively matching noiseless MPS simulations and revealing interaction-induced time delays in wavepacket collisions. The demonstrated approach provides a scalable pathway toward simulating high-energy inelastic scattering on quantum hardware, with potential applicability to more complex theories, provided improvements in optimization, wavepacket size, and qudit resources are pursued.

Abstract

Simulations of collisions of fundamental particles on a quantum computer are expected to have an exponential advantage over classical methods and promise to enhance searches for new physics. Furthermore, scattering in scalar field theory has been shown to be BQP-complete, making it a representative problem for which quantum computation is efficient. As a step toward large-scale quantum simulations of collision processes, scattering of wavepackets in one-dimensional scalar field theory is simulated using 120 qubits of IBM's Heron superconducting quantum computer ibm_fez. Variational circuits compressing vacuum preparation, wavepacket initialization, and time evolution are determined using classical resources. By leveraging physical properties of states in the theory, such as symmetries and locality, the variational quantum algorithm constructs scalable circuits that can be used to simulate arbitrarily-large system sizes. A new strategy is introduced to mitigate errors in quantum simulations, which enables the extraction of meaningful results from circuits with up to 4924 two-qubit gates and two-qubit gate depths of 103. The effect of interactions is clearly seen, and is found to be in agreement with classical Matrix Product State simulations. The developments that will be necessary to simulate high-energy inelastic collisions on a quantum computer are discussed.

Figures (16)

• Figure 1: Mapping the one-dimensional lattice scalar field theory to qubits. For a system of $L$ spatial sites, $n_q$ qubits are used to represent the state of the field at each spatial site. The infinite-dimensional bosonic Hilbert space of $\phi$ at each site is truncated to $2^{n_q}$ values. A maximum magnitude of the field $\phi_\text{max}$ is chosen, and the values of the field in increments of $\delta_\phi$ are assigned to the possible qubit states.
• Figure 2: The algorithm to create variational circuits that can be scaled up to arbitrary system sizes Farrell:2023fgd. A parametrized ansatz with a scalable structure is chosen by considering symmetries of the system and properties specific to the target unitary. The infidelity between the target state and the produced state is minimized to optimize the variational circuit. This process is repeated for a series of increasing system sizes. The parameters are then extrapolated to an arbitrary system size of interest.
• Figure 3: The circuit elements that are used for state preparation and time evolution in simulations of scattering. (a): The circuit elements used to prepare the vacuum. The input state to SC-ADAPT-VQE is a real, symmetric tensor product state over the spatial sites prepared by a single-site variational operator (blue). The circuit implementing $e^{-i\theta O_1}$ (red) is used to couple neighboring sites. The symmetric version of the QFT defined in Ref. Klco:2018zqz, SQFT, is used to account for the symmetric digitization of $\phi$ and $\Pi$ around 0. (b): The translationally-invariant circuit to prepare the vacuum. One layer of SC-ADAPT-VQE is Trotterized into separate terms coupling even-odd and odd-even spatial sites. This circuit has depth 25 and 3 variational parameters (one for the input state, and one for each application of $O_1$). (c): One layer of the brickwall circuit that is used to create particle wavepackets with negative momentum (indicated by the up arrow) on top of the vacuum. To create the corresponding positive-momentum particles, the circuit is flipped and site-wise SWAP gates are added. Each layer has 20 parameters and is depth 2. The building block of this circuit is the variational gate introduced in Ref. Madden:2021dax. (d): The circuit elements that are used to implement time evolution. Both single- (cyan) and two-site (yellow) operators are used, mimicking the terms present in the present in the Hamiltonian of Eq. \ref{['eq:lattice_h']}. The building block of these circuits is the same as (c). (e): Two layers of the translationally-invariant circuit used to implement time evolution. Each layer has depth 2 and 12 parameters. Single- and two-site operators are applied in an iterative fashion.
• Figure 4: Convergence of the wavefunction and state preparation parameters in the vacuum preparation circuit shown in Fig. \ref{['fig:circuit_elements']}b for both the free and interacting theory. (a): $I_4$ taken over a region of four contiguous lattice sites (eight qubits) as a function of system size. One layer of SC-ADAPT-VQE is used. (b): $I_4$ as a function of number of SC-ADAPT-VQE layers used in the ansatz for a $L=8$ system. Each layer uses a longer-range operator $O_d$ given in Eq. \ref{['eq:pi_phi_d']} to create correlations that span more lattice sites. (c): An example of the exponential convergence of one of the angles parameterizing the $\lambda=0$ vacuum preparation circuit with one layer of SC-ADAPT-VQE, as a function of system size. The remainder of the angles are given numerically in App. \ref{['sec:variational_params']}.
• Figure 5: (a): The dispersion relation $E_k$ and (b): the group velocity $v_k$ for $m=1/2$ and $\lambda=0,2$. Exact computation is possible in the continuum (faint solid lines) and in the free lattice theory (dashed lines). The values of $E_k$ and $v_k$ are computed numerically by using ED on the Hamiltonian of Eq. \ref{['eq:lattice_h']} and projecting onto each momentum sectors (solid lines with circle and square markers). The numerical computations are done on a lattice of $L=10$ spatial sites with $n_q=2$ qubits per site. Discrete gradients are used for the numerical computations of $v_k$. (c): The convergence of the prepared particle wavepackets occupying three spatial sites with momentum $k=-\pi/3$, as a function of number of layers of the circuit from Fig. \ref{['fig:circuit_elements']}c. The local infidelity spanning the width of the wavepacket, $I_3$, is used to measure the quality of the prepared wavefunction for $\lambda=0,2$.